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College
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College Mathematics
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2024
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1. y= 5x 2. x=5y 3. 5y/5 = x/5 y= x/5 4.f*-1(x)=1/5x 1.y = x - 7 2.x= y -7 1.f*-1(x)=1/ 3.y=x+7 4.f*-1(x)=x +7 5x 2.x=1/5y 3.y=5x 1.y = x-8/ 3 2.x=y-8/ 3 3)3x=y-8 4.y=3x+8 5.f*-1(x)= 3x+8 1.y=-2/5x+ 1 2.x=-2/5y + 1 3.y=-5/2(x-1) 4.f*-1(x)=-5/2x + 5/2 1.y= (x+4)*3 - 6 2.x=(y+4)*3 - 6 1.y=x*2-9 2.x=y*2-9 3. 4. 3.y*2=x + 9 4.f*-1(x)= x + 9 f x-8 = x-8 + 8 f(x)= x^2 + 8 f g (x) = f(x) = -4x+5 x-8 = x - 8 + 8 = x+0 = x x-8 Inverse Functions Worksheet Inverse functions are a fundamental concept in algebra that allow us to "undo" the effects of a function. For linear and quadratic functions, we can find the inverse both algebraically and graphically. Two functions are inverses of each other if they "undo" each other's operations, resulting in the original input. Fill in the Blank: Fill in the blank with the correct words. 1. To find the inverse of a function algebraically, we first replace f(x) with y ______________. swap 2. The next step is to ______________ the sides of the equation. new) y 3. We then solve for ______________ to get the inverse function. 4. Graphically, inverse functions are reflections of each other over the line y=x ______________. (b,a) 5. If (a,b) is a point on the graph of f(x), then ______________ is a point on the graph of f^(-1)(x). Word bank: y, x, swap, y = x, (b,a) Multiple Choice Questions: Choose the correct answer from the choices for each question. 1. Which of the following is true for the graphs of a function and its inverse? a) They are parallel b) They are perpendicular c) They are reflections over y = x d) They are translations of each other 2. If f(x) = 2x + 3, what is f^(-1)(x)? a) (x - 3)/2 b) (x + 3)/2 c) 2x - 3 d) (3 - x)/2 3. What is the domain of f^(-1)(x) if f(x) = x^2 and the domain of f(x) is all real numbers? a) All real numbers b) All non-negative real numbers c) All negative real numbers d) [0, ∞) 4. If f(g(x)) = x for all x in the domain of g, and g(f(x)) = x for all x in the domain of f, then: a) f(x) and g(x) are linear functions b) f(x) and g(x) are quadratic functions c) f(x) and g(x) are inverse functions d) f(x) and g(x) are the same function 5. Which of the following functions is its own inverse? a) f(x) = x + 1 b) f(x) = x^2 c) f(x) = 1/x, x ≠ 0 d) f(x) = x^3 Open Ended Questions: Answer the following questions in complete sentences: 1. Explain the process of finding the inverse of a quadratic function algebraically. What additional consideration must be made compared to finding the inverse of a linear function? To find the inverse of a quadratic function algebraically, you first replace f(x) with y, then swap x and y, and solve for y. However, unlike linear functions, quadratic functions are not one-to-one over their entire domain, so you must restrict the domain to ensure the inverse exists. This involves considering the vertex of the parabola and choosing either the left or right side of the vertex as the domain for the inverse function. 1. How can you determine if two functions are inverses of each other graphically? Describe the visual relationship between their graphs. If they plot symmetrically over the y=x identity graph y=x (6,-1),(2,-7),(-3,-5) they are mirror images (-2,0) (2,0)